# contextual_active_model_selection__144b0525.pdf

Contextual Active Model Selection

Xuefeng Liu1 , Fangfang Xia2, Rick L. Stevens1,2, Yuxin Chen1

1Department of Computer Science, University of Chicago 2Argonne National Laboratory

While training models and labeling data are resource-intensive, a wealth of pretrained models and unlabeled data exists. To effectively utilize these resources, we present an approach to actively select pre-trained models while minimizing labeling costs. We frame this as an online contextual active model selection problem: At each round, the learner receives an unlabeled data point as a context. The objective is to adaptively select the best model to make a prediction while limiting label requests. To tackle this problem, we propose CAMS, a contextual active model selection algorithm that relies on two novel components: (1) a contextual model selection mechanism, which leverages context information to make informed decisions about which model is likely to perform best for a given context, and (2) an active query component, which strategically chooses when to request labels for data points, minimizing the overall labeling cost. We provide rigorous theoretical analysis for the regret and query complexity under both adversarial and stochastic settings. Furthermore, we demonstrate the effectiveness of our algorithm on a diverse collection of benchmark classification tasks. Notably, CAMS requires substantially less labeling effort (less than 10%) compared to existing methods on CIFAR10 and DRIFT benchmarks, while achieving similar or better accuracy.

1 Introduction

As pre-trained models become increasingly prevalent in a variety of real-world machine learning applications [2, 11, 53], there is a growing demand for label-efficient approaches for model selection, especially when facing varying data distributions and contexts at run time. Oftentimes, no single pre-trained model achieves the best performance for every context, and a proper approach is to construct a policy for adaptively selecting models for specific contexts [48]. For instance, in medical diagnosis and drug discovery, accurate predictions are of paramount importance. The diagnosis of diseases through pathologist or the determination of compound chemical properties through lab testing can be costly and time-consuming. Different models may excel in analyzing different types of pathological images [1, 3, 23] or chemical compounds [17, 32, 46]. Furthermore, in many real-world applications, the collection of labels for model evaluation can be expensive and data instances may arrive as a stream rather than all at once. This scenario necessitates cost-effective and robust online algorithms capable of determining the most efficient model selection policy even when faced with a limited supply of labels, a scenario not fully addressed by previous works that typically assume access to all labels [6, 7, 27, 54].

Recently, the problem of online model selection with the consideration of label acquisition costs was studied in a context-free setting by Karimi et al. [39]. However, this approach doesn t fully capture the dynamics of data contexts that are essential in many applications. Recognizing this gap, in this paper, we consider a more general problem setting that incorporates context information for adaptive model selection. We introduce CAMS, an algorithm for active model selection that dynamically adapts to the data context to choose the most suitable models for an arbitrary data stream. As highlighted in

Correspondence to: Xuefeng Liu <xuefeng@uchicago.edu>.

38th Conference on Neural Information Processing Systems (Neur IPS 2024).

Setup Algorithm Online bagging* Hedge EXP3 EXP4 Query by Committee Model Picker CAMS [54] [27] [7] [7] [65] [39] (ours) bagging online learning bandit contextual bandits active learning model selection (ours) model selection full-information active queries context-aware

We regard arms as models when comparing CAMS against bandit algorithms, such as EXP3/EXP4. * Online ensemble learning aims to build a composite model by aggregating multiple models rather than selecting the best model (for a given context).

Table 1: Comparing CAMS against related work in terms of problem setup.

Table 1, CAMS aims to address the need for adaptive and effective model selection, by bridging the gap between contextual bandits, online learning, and active learning.

Our key contributions are summarized as follows: We investigate a novel problem which we refer to as contextual active model selection, and introduce a novel principled algorithm that features two key technical components: (1) a contextual online model selection procedure, designed to handle both stochastic and adversarial settings, and (2) an active query strategy. The proposed algorithm is designed to be robust to heterogeneous data streams, accommodating both stochastic and adversarial online data streaming scenarios.

We provide rigorous theoretical analysis on the regret and query complexity of the proposed algorithms. We establish regret upper bounds for both adversarial and stochastic data streams under limited label costs. Our regret upper bounds are within constant factors of the existing lower bounds for online learning problems with expert advice under the full information setting.

Empirically, we demonstrate the effectiveness and robustness of our approach on a variety of online model selection tasks spanning different application domains (from generic ML benchmarks such as CIFAR10 to domain-specific tasks in biomedical analysis), data scales (ranging from 80 to 10K), data modalities (i.e., tabular, image, and graph-based data), and label types (binary or multiclass labels). For the tasks evaluated, (1) CAMS outperforms all competing baselines by a significant margin. (2) Asymptotically, CAMS performs no worse than the best single model. (3) CAMS is not only robust to adversarial data streams but also can efficiently recover from malicious experts (i.e. inferior pre-trained models).

2 Related Work

Contextual bandits. Classical bandit algorithms (e.g., [6, 7]) aim to find the best arm(s) through a sequence of actions. When side information (e.g., user profile for recommender systems or environmental context for experimental design) is available, many bandit algorithms can be lifted to the contextual setting: For example, EXP4 [7, 9, 52] considers the bandit setting with expert advice: At each round, experts announce their predictions of which actions are the most promising for the given context, and the goal is to construct a expect selection policy that competes with the best expert from hindsight. In bandit problems, the learner only gets to observe the reward for each action taken. In contrast, for the online model selection problem considered in this work where an action corresponds to choosing a model to make prediction on an incoming data point we get to see the loss/reward of all models on the labeled data point. By utilizing the information from unchosen arms, it could significantly reduce the cumulative regret. In this regard, this work aligns more closely with online learning with full information setting, where the learner has access to the loss of all the arms at each round (e.g. as considered in the Hedge algorithm [13, 14, 27, 36]).

Online learning with full information. A clear distinction between our work and online learning is that we assume the labels of the online data stream are not readily available but can be acquired at each round with a cost. In addition, the learner only observes the loss incurred by all models on a data point when it decides to query its label. In contrast, in the canonical online learning setting, labels arrive with the data and one gets to observe the loss of all candidate models at each round. Similar setting also applies to other online learning problems, such as online boosting or bagging. A related work to ours is online learning with label-efficient prediction [16], which proposes an online learning algorithm with matching upper and lower bounds on the regret. However, they consider a fixed query probability that leads to a linear query complexity. Our algorithm, inspired by uncertainty sampling in active learning, achieves an improved query complexity with the adaptive query strategy while maintaining a comparable regret.

Stream-based Active learning. Active learning aims to achieve a target learning performance with fewer training examples [64]. The active learning framework closest to our setting is queryby-committee (QBC) [65], in particular under the stream-based setting [35, 47]. QBC maintains a committee of hypotheses; each committee member votes on the label of an instance, and the instances with the maximal disagreement among the committee are considered the most informative labels. Note that existing stream-based QBC algorithms are designed and analyzed assuming i.i.d. data streams. In comparison, our work uses a different query strategy as well as a novel model recommendation strategy, which also applies to the adversarial setting.

Active model selection. Active model selection captures a broad class of problems where model evaluations are expensive, either due to (1) the cost of evaluating (or probing ) a model, or (2) the cost of annotating a training example. Existing works under the former setting [49, 59] and online setting [21, 66] often ignore context information and data annotation cost, and only consider partial feedback on the models being evaluated/ probed on i.i.d. data. The goal is to identify the best model with as few model probes as possible. This is quite different from our problem setting which considers the full information setting as well as non-negligible data annotation cost. [71] proposes that the optimal model choice is influenced by the sample size rather than any individual sample feature. [44] addresses the active model selection problem, however both works do not adopt a stream-based approach. For the later, apart from Karimi et al. [39], an online contextual-free model selection work, as shown in Table 1, most existing works assume a pool-based setting where the learner can choose among the pool of unlabeled data [4, 29, 43, 49, 60, 61, 68, 76], and the goal is to identify the best model with a minimal set of labels.

3 Problem Statement

Notations. Let X be the input domain and Y := {0, . . . , c 1} be the set of c possible class labels for each input instance. Let F = {f1, . . . , fk} be a set of k pre-trained classifiers over X Y. A model selection policy π : X k 1 maps any input instance x X to a distribution over the pre-trained classifiers F, specifying the probability π (x) of selecting each classifier under input x. Here, k 1 denotes the k-dimensional probability simplex w Rk : |w| = 1, w 0 . One can interpret a policy π as an expert that suggests which model to select for a given context x.

Let Π be a collection of model selection policies. In this paper, we propose an extended policy set Π := Π {πconst 1 , . . . , πconst k } which also includes constant policies that always suggest a fixed model. Here, πconst j ( ) := ej, and ej k 1 denotes the canonical basis vector with ej = 1. Unless otherwise specified, we assume Π is finite with |Π| = n, and |Π | n + k. As a special case, when Π = , our problem reduces to the contextual-free setting.

The contextual active model selection protocol. Assume that the learner knows the set of classifiers F as well as the set of model selection policies Π. At round t, the learner receives a data instance xt X as the context for the current round, and computes the predicted label ˆyt,j = fj (xt) for each pre-trained classifier indexed by j [k]. Denote the vector of predicted labels by all k models by ˆyt := [ˆyt,1, . . . , ˆyt,k] . Based on previous observations, the learner identifies a model/classifier fjt and makes a prediction ˆyt,jt for the instance xt. Meanwhile, the learner can obtain the true label yt only if it decides to query xt. Upon observing yt, the learner incurs a query cost, and receives a (full) loss vector ℓt = I{ˆyt =yt}, where the jth entry ℓt,j := I{ˆyt,j =yt} corresponds to the 0-1 loss for model j [k] at round t. The learner can then use the queried labels to adjust its model selection criterion for future rounds.

Performance metric. If xt is misclassified by the model jt selected by learner at round t, i.e. ˆyt,jt = yt, it will be counted towards the cumulative loss of the learner, regardless of the learner making a query. Otherwise, no loss will be incurred for that round. For a learning algorithm A, its cumulative loss over T rounds is defined as LA T := PT t=1 ℓt,jt.

In practice, the choice of model jt at round t by the learner A could be random: For stochastic data streams where (x, y) arrives i.i.d., the learner may choose different models for different random realizations of (xt, yt). For the adversarial setting where the data stream {(xt, yt)}t 1 is chosen by an adversary before each round, the learner may randomize its choice of model to avoid a constant loss at each round [33]. Therefore, due to the randomness of LA T , we consider the expected cumulative loss E[LA T ] as a key performance measure of the learner A. To characterize the progress of A, we

consider the regret formally defined as follows as the difference between the cumulative loss received by the learner and the loss if the learner selects the best policy π Π in hindsight.

For stochastic data streams, we assume that each policy i recommends the most probable model w.r.t. πi(xt) for context xt. We use maxind(w) := arg maxj:wj w wj to denote the index of the maximal-

value entry2 of w. Since (x, y) are drawn i.i.d., we define µi = 1

T PT t=1 Ext,yt ℓt,maxind(πi(xt)) . This leads to the pseudo-regret for the stochastic setting over T rounds, defined as

RT (A) = E[LA T ] T min i [|Π |] µi. (1)

In an adversarial setting, since the data stream (and hence the loss vector) is determined by an adversary, we consider the reference best policy to be the one that minimizes the loss on the adversarial data stream, and the expected regret

RT (A) = E[LA T ] min i [|Π |]

t=1 ℓt,i, (2)

where ℓt,i := πi (xt) ,ℓt denotes the expected loss if the learner commits to policy πi, randomizes and selects jt πi (xt) (and receives loss ℓt,jt) at round t. Our goal is to devise a principled online active model selection strategy to minimize the regret as defined in (1) or (2), while maintaining a low total query cost. For convenience, we refer the readers to App. B for a summary of the notations used in this paper.

4 Contextual Active Model Selection

In this section, we introduce our main algorithm for both stochastic and adversarial data streams.

1: Input: Models F, policies Π, #rounds T, budget b 2: Initialize loss L0 0; query cost C0 0 3: Set Π Π {πconst 1 , . . . , πconst k } according to Eq. (3) 4: for t = 1, 2, ..., T do 5: Receive xt 6: ηt SETRATE(t, xt, |Π |)

7: Set qt,i exp ηt Lt 1,i i |Π |

8: jt RECOMMEND(xt, qt) 9: Output ˆyt,jt ft,jt as the prediction for xt 10: Compute zt in Eq. (4) 11: Sample Ut Ber (zt) 12: if Ut = 1 and Ct b then 13: Query the label yt 14: Ct Ct 1 + 1 15: Compute ℓt: ℓt,j = I {ˆyt,j = yt} , j [|F|]

16: Estimate model loss: ˆℓt,j = ℓt,j

zt , j [|F|]

17: Update ℓt: ℓt,i πi(xt), ˆℓt,j , i [|Π |] 18: Lt = Lt 1 + ℓt 19: else 20: Lt = Lt 1 21: Ct Ct 1 22: end if 23: end for

21: procedure SETRATE(t, xt, m) 22: if STOCHASTIC then

t 24: end if 25: if ADVERSARIAL then 26: Set ρt as in adversarial setting section

t + ρt c2 ln c q

T 28: end if 29: return ηt 30: end procedure

29: procedure RECOMMEND(xt, qt) 30: if STOCHASTIC then 31: wt = P

i |Π | qt,iπi(xt) 32: jt maxind(wt) 33: end if 34: if ADVERSARIAL then 35: it qt 36: jt πit (xt) 37: end if 38: return jt 39: end procedure

Figure 1: The Contextual Active Model Selection (CAMS) algorithm

Contextual model selection. Our key insight underlying the contextual model selection strategy extends from the online learning with expert advice framework [13, 27]. We start by appending the constant policies that always pick single pre-trained models to form the extended policy set Π (Line 3, in Fig. 1). This allows CAMS to be at least as competitive as the best model. Then, at each round, CAMS maintains a probability distribution over the (extended) policy set Π , and updates those according to the observed loss for each policy. We use qt := (qt,i)i |Π | to denote the probability distribution over Π at t. Specifically, the probability qt,i is computed based on

the exponentially weighted cumulative loss, i.e. qt,i exp ηt Lt 1,i where Lt,i := Pt τ=1 ℓτ,i denotes the cumulative loss of policy i.

2Assume ties are broken randomly.

For adversarial data streams, it is natural for both the online learner and the model selection policies to randomize their actions to avoid linear regret [33]. Following this insight, CAMS randomly samples a policy it qt, and based on the current context xt samples a classifier jt πit (xt) to recommend at round t.

Under the stochastic setting, CAMS adopts a weighted majority strategy [45] when selecting models. The vector of each model s weighted votes from the policies, wt = P i |Π | qt,iπi(xt), is interpreted as a distribution induced by the weighted policy. The model jt = maxind(wt) which receives the highest probability becomes the recommended model at round t. This deterministic model selection strategy is commonly used in stochastic online optimization [33]. An alternative strategy is to take a randomized approach as in the adversarial setting, or take a Follow-the-Leader approach [42] and go with the most probable model recommended by the most probable policy (i.e. use wt = πmaxind(qt)(xt)).As shown in experimental results section and further discussed in Appendix (outperformance over the best policy/expert section), CAMS outperforms these policies in a wide range of practical applications. The model selection steps are detailed in Line 5-9 in Fig. 1.

Active queries. Under a limited budget, we intend to query the labels of those instances that exhibit significant disagreement among the pre-trained models F. To achieve this goal, we design an adaptive query strategy with query probability zt. Concretely, given context xt, model predictions ˆyt and model distribution wt, we denote by ℓy t := wt, I {ˆyt = y} as the expected loss if the true label is y. We characterize the model disagreement as

E (ˆyt, wt) := 1

y Y, ℓy t (0,1)

ℓy t logc 1 ℓy t . (3)

Intuitively, when ℓy t is close to 0 or 1, there is little disagreement among the models in labeling xt as y, otherwise there is significant disagreement. We capture this insight with function h(x) = x log x. Since the label yt is unknown upfront when receiving xt, we iterate through all the possible labels y Y and take the average value as in Eq. (3). Note that E takes a similar algebraic form to the entropy function, although it does not inherit the information-theoretic interpretation.

With the model disagreement term defined above, we consider an adaptive query probability3

zt = max δt 0, E (ˆyt, wt) , (4)

where δt 0 = 1

t (0, 1] is an adaptive lower bound on the query probability to encourage exploration at an early stage. The query strategy is summarized in Line 10-14 in Fig. 1.

Model updates. Now define Ut Ber (zt) as a binary query indicator that is sampled from a Bernoulli distribution parametrized by zt. Upon querying the label yt, one can calculate the loss for each model fj F as ℓt,j = I {ˆyt,j = yt}. Since CAMS does not query all the i.i.d. examples, we introduce an unbiased loss estimator for the models, defined as ˆℓt,j = ℓt,j

zt Ut. The unbiased loss of policy πi Π can then be computed as ℓt,i = πi(xt), ˆℓt,j . In the end, CAMS computes the (unbiased) cumulative loss of policy πi as LT,i = PT t=1 ℓt,i, which is used to update the policy probability distribution in next round. Pseudocode for the model update steps is summarized in Line 15-21 in Fig. 1.

Remark. CAMS runs efficiently with time complexity O (nk) per round and space complexity O ((n + k) k). At each round, each model selection policy specifies a probability distribution over the models for the given context. When these distributions correspond to constant Dirac delta distributions (regardless of the context), the problem reduces to the context-free problem investigated by Karimi et al. [39].

5 Theoretical Analysis

We now present theoretical bounds on the regret (defined in Eq. (1) and Eq. (2), respectively) and the query complexity of CAMS for both the stochastic and the adversarial settings.

3For convenience of discussion, we assume that those rounds where all policies in Π select the same models or all models F make the same predictions are removed as a precondition.

5.1 Stochastic setting

Under the stochastic setting, the cumulative loss of CAMS over T rounds as specified by the RECOMMEND procedure is LCAMS T = PT t=1 ˆℓt,maxind(wt) where recall wt = P

i |Π | qt,iπi(xt) is the probability distribution over F induced by the weighted policy.

Let i = arg mini [|Π |] µi be the index of the best policy (µi denotes the expected loss of policy i, as defined in problem statement section. The cumulative expected loss of policy i is Tµi ; therefore the expected pseudo-regret (Eq. (1)) is RT (CAMS) = E h PT t=1 ˆℓt,maxind(wt) i Tµi .

Define := mini =i (µi µi ) as the minimal sub-optimality gap4 in terms of the expected loss against the best policy i . Furthermore, let wt i := πi (xt) be probability distribution over F induced by policy i at round t. We define γ := minxt{maxwj wt i wj maxwj wt i ,j =maxind(wt i ) wj} (5) as the minimal probability gap between the most probable model and the rest (assuming no ties) induced by the best policy i . We further define b = pmin logc (1/pmin), where pmin = mins,i π(xs) denotes the minimal model selection probability by any policy5. As our first main theoretical result, we show that, without exhaustively querying the labels of the stochastic stream, CAMS achieves constant expected regret.

Theorem 1. (Regret) In the stochastic environment, with probability at least 1 δ, CAMS achieves

constant expected pseudo regret RT (CAMS) = ln |Π | 1

ln |Π | 2b2 ln 2

Note that in the stochastic setting, a lower bound of Ω((log Π ) / ) was shown in Mourtada and Gaïffas [50] for online learning problems with expert advice under the full information setting (i.e. assuming labels are given for all data points in the stochastic stream). To establish the proof of Theorem 1, we consider a novel procedure to connect the weighted policy by CAMS to the best policy πi . Conceptually, we would like to show that, after a constant number of rounds τconst, with high probability, the model selected by CAMS (Line 32) will be the same as the one selected by the best policy i . In that way, the expected pseudo regret will be dominated by the maximal cumulative loss up to τconst. Toward this goal, we first bound the weight of the best policy wt,i as a function of t, by choosing a proper learning rate ηt (CAMS, Line 23). Then, we identify a constant threshold τconst, beyond which CAMS exhibits the same behavior as πi with high probability. Finally, we obtain the regret bound by inspecting the regret at the two stages separately. The formal statement of Theorem 1 and the detailed proof are deferred to App. E.1.

Next, we provide an upper bound on the query complexity in the stochastic setting. Theorem 2. (Query Complexity). For c-class classification problems, with probability at least 1 δ, the expected number of queries made by CAMS over T rounds is upper bounded by ln |Π | 1

ln |Π | 2b2 ln 2

ln T c ln c.

Theorem 2 is built upon Theorem 1, where the the key idea behind the proof is to relate the number of updates to the regret. When Tµi , LT, are regarded as constants (given by an oracle), the query-complexity bound is then sub-linear w.r.t. T. Note that the number of class labels c affects the quality of the query complexity bound. The intuition behind this result is, with larger number of classes, each query may carry more information upon observation. For instance, in an extreme case where only one expert always recommends the best model and others gives random recommendations of models (and predicts random labels), having more classes lowers the chance of a model making the correct guess, and therefore helps to "filter out" those suboptimal experts in fewer rounds hence being more query efficient. We defer the proof of Theorem 2 to App. E.2.

5.2 Adversarial setting

Now we consider the adversarial setting. Let LT, := mini [|Π |] PT t=1 ℓt,i be the cumulative loss of the best policy. The expected regret (Eq. (2)) for CAMS equals to RT (CAMS) =

4w.l.o.g. assume there is a single best policy, and thus > 0. 5We assume pmin > 0 per the policy regularization criterion in Appendix C.3. (cf. Algorithm 1 on Regularized policy π(xt)) .

Cumulative loss

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(a) CIFAR10

0 250 500 750 1000 1250 1500 1750 2000 Query cost

0 10 20 30 40 50 60 70 80 Query cost

(c) VERTEBRAL

0 250 500 750 1000 1250 1500 1750 2000 Query cost

Figure 2: Main results. Comparison of CAMS with 7 baselines across 4 diverse benchmarks in terms of cost effectiveness. We plot the cumulative loss as we increase the query cost for a fixed number of rounds T and maximal query cost B (from left to right: T = 10000, 3000, 80, 4000, and B = 1200, 2000, 80, 2000). CAMS outperforms all baselines. Algorithms: 4 contextual {Oracle, CQBC, CIWAL, CAMS} and 4 non-contextual baselines {RS, QBC, IWAL, MP} are included (see Section ). 90% confident interval are indicated in shades.

E PT t=1 qt, ℓt LT, . We show that under the adversarial setting, CAMS achieves sub-linear regret in T without accessing all labels. Theorem 3. (Regret) Let c be the number of classes and ρt be specified as Line 26-27 in the SETRATE procedure. Under the adversarial setting, the expected regret of CAMS is bounded by

ln c/max{ρT , p

T log |Π |.

The proof is provided in App. F.1. Assuming ρt to be a constant, our regret upper bound in Theorem 3 matches (up to constants) the lower bound of Ω p

T ln |Π | for online learning problems with expert advice under the full information setting [15, 63] (i.e. assuming labels are given for all data points). Hereby, the decaying learning rate ηt as specified in Line 27 is based on two parameters, where 1/

t corresponds to the lower bound δt 0 on the query probability, and ρt 1 maxτ [t 1] wτ, I {ˆyτ = yτ} (6) is a (data-dependent) term that is chosen to reduce the impact of the randomized query strategy on the regret bound (especially when t is large). Intuitively, ρt relates to the skewness of the policy where the max term corresponds to the maximal probability of most probable mispredicted label over t rounds. Note that in theory ρt can be small (e.g. CAMS may choose a constant policy πconst i Π that mispredict the label for xt, which leads to ρt = 0); in such cases, our result still translates to a sublinear regret bound of O(c log c T 3 4 p

log |Π |). Furthermore, in practice, we consider to regularize the policies (App. D.4) to ensure that probability a policy selecting any model is bounded away from 0.

Finally, the following theorem (proof in App. F.2) establishes a query complexity bound of CAMS. Theorem 4. (Query Complexity). Under the adversarial setting, the expected query complexity over

T rounds is O ln T r

1/T } + LT,

6 Experiments

Datasets. We evaluate our approach using five datasets: (1) CIFAR10 [41] contains 60,000 images from 10 different balanced classes. (2) DRIFT [73] is a tabular dataset with 128-dimensional features, based on 13,910 chemical sensor measurements of 6 types of gases at various concentration levels. (3) VERTEBRAL [5] is a biomedical tabular dataset which classifies 310 patients into three classes (Normal, Spondylolisthesis, Disk Hernia) based on 6 attributes. (4) HIV [74] contains over 40,000 compounds annotated with molecular graph features and binary labels (active, inactive) indicating their ability to inhibit HIV replication. (5) Cov Type [24] has 580K samples and contains details including slope, aspect, elevation, measurements of area, and type of forest cover.

Policy sets. We construct the policy sets Π for each dataset following a procedure similar to Meta-selector [48]. In this approach, a set of recommender algorithms is considered, and Meta-selector assigns varying ratings to these algorithms based on the specific user. Concretely, we first construct a set of models trained on different subsamples from each dataset. We then construct a set of policies, which include malicious, normal, random, and biased policy types for each dataset

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Cumulative loss

variance random CAMS

(a) Query strategy ablation study

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Number of queries

RS QBC IWAL MP CQBC CIWAL CAMS

(b) Query complexity study

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MP CAMS best classifier suboptimal classifiers

(c) Context-free environment

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best classifier adversarial policies CAMS

(d) Robust in pure adversarial set

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RS Oracle QBC IWAL MP CQBC CIWAL CAMS

(e) Large dataset (COVTYPE)

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RS Oracle QBC IWAL MP CQBC CIWAL CAMS

(f) Adjust prob. (VERTEBRAL)

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RS Oracle QBC IWAL MP CQBC CIWAL CAMS

(g) Adjust probability (HIV)

0 100 200 300 400 500 600 700 800 Query cost

Cumulative loss

best policy CAMS

(h) CAMS vs best policy (HIV)

Figure 3: Ablation studies. (a) Comparing three query strategies {CAMS, variance-based, random} under same model selection policy. (b) Comparing the increasing rate of CAMS query cost over other baselines. (c) Comparing CAMS with MP in context-free environment. (d) Evaluating the performance of CAMS under a pure adversarial setting. (e) Large dataset. (f,g) Adjustable query probability. (h) CAMS outperforms the best single policy. The ablation study (a)-(d) is conducted on CIFAR10. For additional results on other benchmarks, please refer to the supplemental material.

based on different models and features. Details on the classifiers and policies are provided in the supplemental materials. The malicious policy provides contrary advice; the random policy provides random advice; the biased policy provides biased advice by training on a biased distribution for classifying specific classes. The normal policy gives reasonable advice, being trained under a standard process on the training set. We represent the output of the ith policy as πi (xt), indicating the rewards distribution of all the base classifiers on xt. In total, we create 80, 10, 6, 4 classifiers and 85, 11, 17, 20 policies for CIFAR10, DRIFT, VERTEBRAL, and HIV, respectively.

Baselines. We evaluate CAMS against both contextual and non-contextual active model selection baselines. We consider four non-contextual baselines: (1) Random Query Strategy (RS) which queries the instance label with a fixed probability b

T ; (2) Model Picker (MP) [39] that employs variancebased active sampling with a coin-flip query probability max {v (ˆyt, wt) , ηt}, where the variance term is defined as v (ˆyt, wt) = maxy Y ℓy t 1 ℓy t ; (3) Query by Committee (QBC) implementing committee-based sampling [22]; and (4) Importance Weighted Active Learning (IWAL) [8] that calculates query probability based on labeling disagreements of surviving classifiers. Since no contextual baselines exist yet, we propose contextual versions of QBC and IWAL as (5) CQBC and (6) CIWAL. Both extensions maintain their respective original query strategies but incorporate the context into the cumulative rewards. For model selection, CAMS, MP, CQBC, and CIWAL recommend the classifier with the highest probability. The other baselines use Follow-the-Leader (FTL), recommending the model with the minimum cumulative loss for past queried instances. Finally, we add (7) Oracle to represent the best single policy with the minimum cumulative loss, with the same query strategy as CAMS.

6.1 Main results

Fig. 2 visualizes the cost effectiveness of CAMS and the baselines. Here, we define cost effectiveness as the measure of how quickly the cumulative loss decreases in response to an increase in query cost. Fig. 2 demonstrates that CAMS outperforms all the comparison methods across all benchmarks. Remarkably, it outperforms even the oracle on the VERTEBRAL (Fig. 2c) and HIV (Fig. 2d) benchmarks with fewer than 10 and 20 queries, respectively. In the case of the VERTEBRAL benchmark, CAMS outperforms the best baseline in query cost by a margin of 20%, despite the fact that 11 out of the 17 experts provided malicious or random advice. This level of performance is attained by utilizing an active query strategy to retrieve highly informative data, thereby maximizing the differentiation between models and policies within the constraints of a limited budget. Additionally, the model selection strategy allows for effectively combining the expertise among the experts.

6.2 Ablation studies

Effectiveness of active querying. In Fig. 3a and Fig. 3b, we perform ablation studies to demonstrate the effectiveness of our active query strategy. We fix the model recommendation strategy as the one used by CAMS, and compare three query strategies: (1) CAMS, (2) the state-of-the-art variancebased query strategy from Model Picker [39] (referred to as variance ), and (3) a random query strategy. Figure 3a demonstrates that CAMS has the fastest convergence rate in terms of cumulative loss on CIFAR10, implying effective use of queried labels. Furthermore, CAMS not only achieves the minimum cumulative loss but also incurs significantly lower query costs, with reductions of 71% and 95% compared to the variance and random strategies respectively as showed in Fig. 3b. This suggests that CAMS selectively queries data to optimize policy improvement, whereas the other strategies may query unnecessary labels, including potentially noisy or uninformative ones, which impede policy improvement and convergence.

Robustness. In Fig. 2, 3c, 3d,3e, 3f, and 3g, CAMS exhibits robustness in a variety of environmental settings. Firstly, As shown in Fig. 2, CAMS outshines other methods in a contextual environment, whereas in Fig. 3c, a non-contextual (no experts) environment, it achieves comparable performance to the state-of-the-art Model Picker in identifying the best classifier. Secondly, CAMS is robust in both stochastic and adversarial environments. As demonstrated in Fig. 2, CAMS surpasses other methods in a stochastic environment. Additionally, as illustrated in Fig. 3d, in a worst-case adversarial environment, CAMS effectively recovers from adversarial actions and approaches the performance of the best classifier (see App. G.5). We further observe that CAMS demonstrates robustness to varying scales of data, where the online stream sizes range from 80 to 10K (Fig. 2) to 100K (Fig. 3e, where we randomly sample 100K samples from the Cov Type dataset [24]).

In Fig. 2, we assume that the stream length T is hidden and not used as input to CAMS. Under the stochastic setting, however, knowing T can provide additional information that one can leverage to optimize the query probability, thereby giving an advantage to some of the baseline algorithms (e.g. random). As an ablation study, in Fig. 3f and Fig. 3g, we assume the stochastic setting where the total length T of the online stream is given. Given the stream length T and query budget b, we may optimize each algorithm by scaling their query probabilities, so that each algorithm allocates its query budget to the top b informative labels in the entire online stream based on its own query criterion. CAMS still ourperform the baselines under the setting.

Improvement over the best classifier and policy. Fig. 3h demonstrates that when provided with good policies, CAMS formulates a stronger policy which incurs no regret. CAMS has the potential to outperform an oracle, especially in rounds where the oracle does not make the optimal recommendation. For instance, in the stochastic version of CAMS (as shown in lines 22-23 and 30-32 of Fig. 1), CAMS recommends a model using a weighted majority vote among all policies, enabling the formation of a new policy in each round by amalgamating the strengths of each sub-optimal policy. This adaptive strategy can potentially outperform any single policy. Moreover, in most real-world scenarios and conducted experiments (as depicted in App. G.6), data streams may not be strictly stochastic, and therefore no single policy consistently performs the best. In such cases, CAMS s weighted policy may find an enhanced combination of advices , leading to improved performance.

7 Conclusion

We introduced CAMS, an online contextual active model selection framework based on a novel model selection and active query strategy. The algorithm was motivated by many real-world use cases that need to make decision by taking both contextual information and the cost into consideration. We have demonstrated CAMS s compelling performance of using the minimum query cost to learn the optimal contextual model selection policy on several diverse online model selection tasks. In addition to the promising empirical performance, we also provided rigorous theoretical guarantees on the regret and query complexity for both stochastic and adversarial settings. We hope our work can inspire future works to handle more complex real-world model selection tasks (e.g. beyond classification or non-uniform loss functions, etc. where our analysis does not readily apply).

Acknowledgements

This work is supported in part by the Rad Bio-AI project (DE-AC02-06CH11357), U.S. Department of Energy Office of Science, Office of Biological and Environment Research, the IMPROVE project under contract (75N91019F00134, 75N91019D00024, 89233218CNA000001, DE-AC02-06CH11357, DE-AC52-07NA27344, DE-AC05-00OR22725), the Laboratory Directed Research and Development (LDRD) funding from Argonne National Laboratory provided by the Director, Office of Science, of the U.S. Department of Energy under Contract No. DE-AC02-06CH11357, the Exascale Computing Project (17-SC-20-SC), a collaborative effort of the U.S. Department of Energy Office of Science and the National Nuclear Security Administration, the University of Chicago Joint Task Force Initiative, the AI-Assisted Hybrid Renewable Energy, Nutrient, and Water Recovery project (DOE DE-EE0009505), and the National Science Foundation under Grant No. IIS 2313131 and IIS 2332475.

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A Impact Statements

This paper introduces a novel framework for adaptive model selection in label-efficient learning. By integrating robust online learning with active query strategies, our algorithm effectively adapts to varying data contexts and minimizes labeling efforts, crucial in domains requiring swift and accurate decisions, such as disease identification and financial predictions. Ethically, the framework s design promotes efficient and context-aware model selection, reducing potential biases associated with context-ignorant model selections. No major ethical concerns are anticipated, given the algorithm s generality and focus on solving practical problems.

B Table of Notations Defined in the Main Paper

notation meaning

Problem Statement X input domain x input instance, x X t, T index of a round, total number of rounds i, j index of policies, models/classifiers n number of policies k number of classifiers Y {0, . . . , c 1}, set of c possible class labels for each input instance c number of class labels, |Y| k 1 k-dimensional probability simplex w Rk : |w| = 1, w 0

f single pre-trained classifier (model) F {f1, . . . , fk}, set of k pre-trained classifiers over X Y π, π(x) model selection policy (expert) π : X k 1, probability of selecting each classifier under input x πconst πconst j ( ) := ej, ej k 1 denotes the canonical basis vector with ej = 1 Π collection of model selection policies Π Π {πconst 1 , . . . , πconst k }, extended policy set including constant policies that always suggest a fixed model |Π|, |Π | n, |Π | (n + k) π π Π , best policy ˆyt,j fj (xt), predicted label for jth pre-trained classifier at round t yt true label of xt ˆyt [ˆyt,1, . . . , ˆyt,k] , predicted labels by all k models at round t ℓt,j I{ˆyt,j =yt}, 0-1 loss for model j [k] at round t ℓt I{ˆyt =yt}, full loss vector upon observing yt A the learner LA T PT t=1 ℓt,jt, cumulative loss over T rounds for a learning algorithm A ℓt,i πi (xt) ,ℓt , expected loss if the learner commits to policy πi and take random selection at round t maxind(w) arg maxj,wj w wj, index of maximal value entry of w

µ 1 T PT t=1 Ext,yt h ˆℓt,maxind(πi(xt)) i

RT (A), RT (A) expected regret in adversarial setting, pseudo-regret for stochastic setting Et[ ] E[ |Ft], Ft = σ E(1),ˆy1, ...,ˆyt 1, E(t)

Algorithm qt (qt,i)i |Π |, probability distribution over Π at t Lt,i Pt τ=1 ℓτ,i, cumulative loss of policy i wt P

i |Π | qt,iπi(xt), distribution induced by the weighted policy ℓy t wt, I {ˆyt = y} , expected loss if the true label is y E (ˆyt, wt) model disagreement function h(x) x log x δt 0 1

t, lower bound of query probability zt max {δt 0, E (ˆyt, wt)}, adaptive query probability ˆℓt,j ℓt,j

zt Ut U query indicator ηt adaptive learning rate ρt 1 maxτ [t 1] wτ, I {ˆyτ = yτ} b query budget bℓ, bℓt,i

i [k] unbiased estimate of classifier loss vector

i [n] unbiased estimate of policy loss vector

b L unbiased cumulative loss of classifiers, policies

Analysis pt,y P

j [k] I {ˆyt,j = y} wj, the total probability of classifiers predicts label y at round t

mini =i i = mini =i (µi µi ), sub-optimality gap γ minxt n maxwj wt i wj maxwj wt i ,j =maxind(wt i ) wj o , sub-optimality model probability gap of πi

i E[eℓ.,i eℓ.,i ], sub-optimality gap or immediate regret of policy i LT, the cumulative loss of oracle at round T

Table 2: Notations used in the main paper

C Summary of Regret and Query Complexity Bounds

We summarize the regret and query complexity bounds (if applicable) of related algorithms in Table 3.

Algorithm Regret Query Complexity

Exp3 [42] 2 Tk log k

Exp3.p [12] 5.15pn T log n

Exp4 [42] 2Tk log n

Exp4.p [9] 6pk T ln n

Model Pickerstochastic [39] 62 maxi ik/ λ2 log k

λ = minj [k]\{i } 2 j/θj 2T log k(1 + 4 c

Model Pickeradversarial [39] 2 2T log k 5 T log k + 2LT,

CAMSSTOCHASTIC

ln |Π | 2b2 ln 2

2 ln |Π | 1

ln |Π | 2b2 ln 2

ln T c ln c

CAMSADVERSARIAL 2c q

ln c/max{ρT , p

T log |Π | O r

1/T } + LT,

Table 3: Regret and query complexity bounds. For the notations in this table: i is the model with the highest expected accuracy; θj = P [ℓ.,j = ℓ.,i ] is the probability that exactly one of j and i correctly classifies a sample; γ and ρT are defined in Eq. (5) and (6), respectively. b = pmin logc (1/pmin), where pmin = mins,i π(xs) denotes the minimal model selection probability by any policy.

Remark 5. When Tµi , LT, are regarded as constants (given by an oracle), the query-complexity bound is then sub-linear w.r.t. T. Remark 6. Note that the number of class labels c affects the quality of the query complexity bound. The intuition behind this result is, with larger number of classes, each query may carry more information upon observation. For instance, in an extreme case where only one expert always recommends the best model and others gives random recommendations of models (and predicts random labels), having more classes lowers the chance of a model making the correct guess, and therefore helps to "filter out" those suboptimal experts in fewer rounds hence being more query efficient. Remark 7. To prove the practical feasibility of CAMS, we have analyzed its time and space complexity. Our analysis shows that CAMS has a time complexity of O (Tnk) in total or O(nk) per round (due to the RECOMMEND procedure under the stochastic setting), and a space complexity of O ((n + k) k). Here, T refers to the online horizon, n denotes the number of policies, and k denotes the number of models. Taking into account these complexities, we can confirm that CAMS is practically feasible.

D Supplemental Materials on Experimental Setup

D.1 Baselines

Model Picker (MP) Model Picker [39] is a context-free online active model selection method inspired by EXP3. Model Picker aims to find the best classifier in hindsight while making a small number of queries. For query strategy, it uses a variance-based active learning sampling method to select the most informative label to query to differentiate a pool of models, where the variance is defined as v (ˆyt, wt) = maxy Y ℓy t 1 ℓy t . The coin-flip query probability is defined as max {v (ˆyt, wt) , ηt} when v (ˆyt, wt) = 0, or 0 otherwise. For model recommendation, it uses an exponential weight algorithm to recommend the model with minimal exponential cumulative loss based on the past queried labels at each round.

Query by Committee (QBC) For query strategy, we have adapted the method of [22] as a disagreement-based selective sampling query strategy for online streaming data. We treat each classifier as a committee member and compute the query probability by measuring disagreement between models for each instance. The query function is coin-flip by vote entropy probability 1 log min (k,|C|) P

k log V (c,x)

k , where V (c, x) stands for the number of committee members assigning a class c for input context x and k is the number of committee. For the model recommendation part, we use the method of Follow-the-Leader (FTL) [42], which greedily recommends the model with the minimum cumulative loss for past queried instances.

Importance Weighted Active Learning (IWAL) We have implemented [8] as the IWAL baseline. For the query strategy part, IWAL computes an adaptive rejection threshold for each instance and assigns an importance weight to each classifier in the hypothesis space Ht. IWAL retains the classifiers in the hypothesis space according to their weighted error versus the current best classifier s weighted error at round t. The query probability is calculated based on labeling disagreements of surviving classifiers through function maxi,j Ht,y [c] ℓ(y) t,i ℓ(y) t,j . For model recommendation, we also adopt the Follow-the-Leader (FTL) strategy.

Random Query Strategy (RS) The RS method queries the label of incoming instances by the coin-flip fixed probability b

T . It also uses the FTL strategy based on queried instances for model recommendation.

Contextual Query by Committee (CQBC) We have created a contextual variant of QBC termed CQBC, which has the same entropy query strategy as the original QBC. For model recommendation, we combine two model selection strategies. The first strategy calculates the cumulative reward of each classifier based on past queries and normalizes it as a probability simplex vector. We also adopt Exp4 s arm recommending vector to use contextual information. Finally, we compute the element-wise product of the two vectors and normalize it to be CQBC s model recommendation vector. At each round, CQBC would recommend the top model based on the classifiers historical performance on queried instances and the online advice matrix for streaming data.

Contextual Importance Weighted Active Learning (CIWAL) We have created a variant version of importance-weighted active learning. Similar to CQBC, CIWAL adopts the query strategy from IWAL and converts the model selection strategy to be contextual. For model selection, we incorporate Exp4 s arm recommendation strategy based on the side-information advice matrix and each classifier s historical performance according to queried instances. We compute the element-wise product of the two vectors as the model selection vector of CIWAL and normalize it as a weighted vector. Finally, CIWAL recommends the classifier with the highest weight.

Oracle: Among all the given policies, oracle represents the best single policy that achieves the minimum cumulative loss, and it has the same query strategy as CAMS.

D.2 Details on policies and classifiers

We constructed different expert-model configurations to reflect the cases in real-world applications6. This section lists the collection of policies and models used in our experiments.

CIFAR10: We have constructed 80 diversified classifiers based on VGG [67], Res Net [34], Dense Net [38], Goog Le Net [69]. We have also used Efficient Net [70], Mobile Nets [37], Reg Net [62], and Res Net to construct 85 diversified policies.

DRIFT: We have constructed ten classifiers using Decision Tree [55], SVM [18], Ada Boost [28], Logistic Regression [20], KNN [19] models. We have also created 8 diversified policies with multilayer perceptron (MLP) models of different layer configurations: (128, 30, 10); (128, 60, 30, 10); (128, 120, 30, 10); (128, 240, 120, 30, 10).

VERTEBRAL: We have built six classifiers using Random Forest [10], Gaussian Process [56], linear discriminant analysis [26], Naive Bayes [31] algorithms. We have constructed policies by using standard scikit-learn built-in models including Random Forest Classifier, Extra Trees Classifier [30], Decision Tree Classifier, Radius Neighbors Classifier [51], Ridge Classifier [57] and K-Nearest Neighbor classifiers.

6To list a few other scenarios beyond the ones used in the paper: In healthcare, models could be the treatments, experts could be the doctors and the context could be the condition of a patient. For any patient (context), doctors (experts) will have their own advice on the treatment (model) recommendation for this patient based on their past experience. In the finance domain, models could be trading strategies, experts could be portfolio managers, and the context could be the stock/equity. Some trading strategies (models) might work well for the information technology sector, and some other models might work well for the energy sector, so depending on the sector of stock (context), different portfolio managers (experts) might have their own advice on different trading strategies (models) based their past trading experience.

HIV: We have used graph convolutional networks (GCN) [40], Graph Attention Networks (GAT) [72], Attentive FP [75], and Random Forest to construct 4 classifiers. We have also used various feature representations of molecules such as MACCS key [25], ECFP2, ECFP4, and ECFP6 [58] molecular fingerprints to build 6 MLP-based policies, respectively.

Cov Type: We have built 6 classifiers using Random Forest, Gaussian Process, linear discriminant analysis, Naive Bayes algorithms. We have constructed 17 policies by using standard scikit-learn built-in models including Random Forest Classifier, Extra Trees Classifier, Decision Tree Classifier, Radius Neighbors Classifier, Ridge Classifier and K-Nearest-Neighbor classifiers.

D.3 Implementation details

We build our evaluation pipeline on top of prior work [39] around the four benchmark datasets. Specifically,

Context xt is the raw context of the data (e.g., the 32x32 image for CIFAR10). Predictions ˆyt contain the predicted label vector of all the classifiers predictions according to the online context xt. Oracle contains the true label yt of xt. Advice matrix contains all policies probability distribution λ over all the classifiers on context xt.

To adapt to an online setting, we sequentially draw random T i.i.d. instances x1:T from the test pool and define it as a realization. For a fair comparison, all algorithms receive data instances in the same order within the same realization.

Algorithm 1 Regularized policy π (xt)

1: Input: context xt, Models F, policy π Π

2: η = P|F| j=1 [π (xt)]j 1 |F| 2

3: return πi(xt)+η

D.4 Regularized policy

As discussed in adversarial section, we wish to ensure that the probability a policy selecting any model is bounded away from 0 so that the regret bound in Theorem 3 is non vacuous. In our experiments, we achieve this goal by applying a regularized policy π as shown in Algorithm 1.

D.5 Summary of datasets and models

We summarize the attributes of datasets, the models, and the model selection policies as follows.

dataset classification total instances test set stream size classifier policy

CIFAR10 10 60000 10000 10000 80 85 DRIFT 6 13910 3060 3000 10 11 VERTEBRAL 3 310 127 80 6 17 HIV 2 40000 4113 4000 4 20 Cov Type 55 580000 100000 100000 6 17

Table 4: Attributes of benchmark datasets

D.6 Hyperparameters

We performed our experiments on a Linux server with 80 Intel(R) Xeon(R) Gold 6148 CPU @ 2.40GHz and total 528 Gigabyte memory.

By considering the resource of server, We set 100 realizations and 3000 stream-size for DRIFT, 20 realizations and 10000 stream-size for CIFAR10, 200 realizations and 4000 stream size for HIV,

300 realization and 80 stream-size for VERTEBRAL. In each realization, we randomly selected stream-size aligned data from testing-set and make it as online streaming data which is the input of each algorithm. Thus, we got independent result for each realization.

A small realization number would increase the variance of the results due to the randomness of stream order. A large realization number would make the result be more stable but at the cost of increasing computational cost (time, memory, etc.). We chose the realization number by balancing both aspects.

E Proofs for the Stochastic Setting

In this section, we focus on the stochastic setting. We first prove the regret bound presented in Theorem 1 and then prove the query complexity presented in Theorem 2 for Algorithm 1.

E.1 Proof of Theorem 1

Before providing the proof of Theorem 1, we first introduce the following lemma. Lemma 8. Fix τ (0, 1). Let qt,i be the probability of the optimal policy i maintained by Algorithm 1 at t, and let b = pmin logc (1/pmin), where pmin = mins,i π(xs) denotes the minimal

model selection probability by any policy7. When t

ln (|Π | 1)τ

, with probability

at least 1 δ, it holds that qt,i τ.

Proof of Lemma 8. W.l.o.g, we assume µ1 µ2 . . . µn+k. Recall that we define =

mini =i i = µ2 µ1 = E[e Lt,2 e Lt,1]

t , and π1 is the policy with the minimal expected loss.

δt ℓt 1,i ℓt 1,1. (7)

where i arg mini =1 Lt 1,i denotes the index of the best empirical policy up to t 1 other than π1. Therefore for i 2, it holds that

e Lt 1,i e Lt 1,i =

We have qt,i = qt,1 = exp( ηt e Lt 1,1) P|Π | i=1 exp( ηt e Lt 1,i) as the weight of optimal expert at round t. Therefore

qt,i = qt,1 = exp ηte Lt 1,1

P|Π | i=1 exp ηte Lt 1,i

(a) = exp ηte Lt 1,1 + ηte Lt 1,i

P|Π | i=1 exp ηte Lt 1,i + ηte Lt 1,i

(b) = exp ηt Pt s=1 δs

exp ηt Pt s=1 δs + P|Π | i=2 exp ηte Lt 1,i + ηte Lt 1,i

exp ηt Pt s=1 δs

exp ηt Pt s=1 δs + |Π | 1

where step (a) is by dividing the cumulative loss of sub-optimal policy πi and step (b) is by the definition of δt in Equation (7).

7We assume pmin > 0 per the policy regularization criterion in Appendix C.3. (cf. Algorithm 1 on Regularized policy π(xt)) .

Let τ (0, 1), such that qt,i exp(ηt Pt s=1 δs) exp(ηt Pt s=1 δs)+|Π | 1 τ. Plugging in ηt = q

t and define

t Pt s=1 δs, we get

t δt + |Π | 1

Therefore, we obtain exp p

t δt (|Π | 1)τ

1 τ . Rearranging the terms, we get

ln (|Π | 1)τ

Next, we seek a high probability upper bound on δt. Denote i µi µ1 for i 1, . . . , |Π |. We know

P(δt 2 ϵ) (a) P(δt i ϵ) = P(1

s=1 δs i ϵ) (b) e tϵ2

Here, step (9a) is by the fact that 2 = mini =1 i i , and step (9b) is by Hoeffding s inequality where b denotes the upper bound on |δs|. Further note that

δs+1 = ℓs,i ℓs,1 = Us

zs πi (xs) π1(xs), I {ˆys = ys}

πi (xs), I {ˆys = ys}

zs Eq. (4) Us πi (xs), I {ˆys = ys} 1 c P

y Y ws, I {ˆys = y} logc 1 ws,I{ˆys =y}

Given pmin = mins,i π(xs), we obtain δs+1 1 pmin logc(1/pmin) and similarly, δs+1

π1(xs),I{ˆys =ys}

zs 1 pmin logc(1/pmin). We hence conclude that |δs+1| b.

2b2 = δ. Therefore, when t ln (|Π | 1)τ

ln |Π |( ϵ)

ln (|Π | 1)τ

that qt,i τ with probability at least 1 δ.

Lemma 9. At round t, when t ln |Π | 1

ln |Π | 2b2 ln 2

2 , it holds that the arm chosen by

the best policy i will be the arm chosen by Algorithm 1 with probability at least 1 δ. That is, arg max n P

i [|Π |] qt,iπi(xt) o = arg max {πi (xt)}.

Proof of Lemma 9. At round t, for Algorithm 1, we have loss Pk j=1 I n j = arg max n P

i [|Π |] qt,iπi(xt) oo bℓt,j. Let qt,i τ. At round t, the best pol-

icy i s top weight arm jt,i s probability max {πi (xt)} is at least 1

k. The second rank probability of πi (xt) is maxj [πi (xt)]j =maxind(πi (xt)). Let us define

γ := min xt

max wj wt i wj max wj wt i ,j =maxind(wt i ) wj

= max {πi (xt)} max j [πi (xt)]j =maxind(πi (xt)) ,

as the minimal gap in model distribution space of best policy. The arm recommended by the best policy i of CAMS will dominate CAMS s selection, when we have

qt,i max {πi (xt)} (1 qt,i ) + qt,i max j [πi (xt)]j =maxind(πi (xt))

Rearranging the terms, and by

qt,i γ Eq. (10) = qt,i max {πi (xt)} max j [πi (xt)]j =maxind(πi (xt))

Therefore, we get τ (γ) (1 τ), and thus τ 1 γ+1.

Set τ 1 γ+1. By Lemma 8, we get

ln |Π | ( ϵ)

ln |Π | ( ϵ)

ln |Π | 2b2

where the last step is by applying 2e tϵ2

2b2 = δ, thus, ϵ = q

δ . Dividing both sides by t

ln |Π | t q

ln |Π | 2b2 ln 2

ln (|Π |) 2b2 ln 2

ln |Π | 2b2 ln 2

So, when t ln |Π | 1

ln |Π | 2b2 ln 2

2 , it holds that arg max n P

i [|Π |] qt,iπi(xt) o =

arg max {πi (xt)}.

Proof of Theorem 1. Therefore, with probability at least 1 δ , we get constant regret ln |Π | 1

ln |Π | 2b2 ln 2

Furthermore, with probability at most δ, the regret is upper bounded by T. Thus, we have

R (T) (1 δ)

ln |Π | 2b2 ln 2

ln |Π | (2 ln T + 2 ln 2) p

where step (a) by setting δ = 1

T , and where γ in Eq. (10) is the min gap.

E.2 Proof of Theorem 2

In this section, we analyze the query complexity of CAMS in the stochastic setting, where we take a similar approach as proposed by Karimi et al. [39] for the context-free model selection problem. Our main idea is to derive from query indicator Ut and query probability zt. We first used Lemma 10 to bound the expected number of queries PT t=1 Ut by the sum of query probability as PT t=1 δt 0 + PT t=1 E (ˆyt, wt). Then we used Lemma 11 to bound the first item (which corresponds to the lower bound of query probability over T rounds) and applied Lemma 12 to bound the second term (which characterizes the model disagreement). Finally, we combined the upper bounds on the two parts to reach the desired result.

Lemma 10. The query complexity of Algorithm 1 is upper bounded by

y Y wt,ℓy t log|Y| 1 wt,ℓy t |Y|

Proof. Now we have model disagreement defined in Eq. (3), the query probability defined in Eq. (4), and the query indicator U. Let us assume, at each round, we have query probability zt > 0, which indicates we will not process the instance that all the models prediction are the same.

At round t, from query probability Eq. (4), we have

zt = max δt 0, E (ˆyt, wt)

δt 0 + E (ˆyt, wt) ,

where the inequality is by applying that A, B 0, max{A, B} A + B.

Thus, in total round T, we could get the following equation as the cumulative query cost,

y Y wt,ℓy t log|Y| 1 wt,ℓy t |Y|

where the inequality is by inputting δt 0 = 1

t and Eq. (3).

Lemma 11. PT t=1 1

Proof. We can bound the LHS as follows:

Lemma 12. Denote the true label at round t by yt, and define pt,y := P

j [k] I {ˆyt,j = y} wj. Further define Rt := P

t 1 pt,yt as the expected cumulative loss of Algorithm 1 at t. Then

y Y wt,ℓy t log|Y| 1 wt,ℓy t |Y| RT log|Y| T 2(|Y| 1)

Proof of Lemma 12. Suppose at round t, the true label is yt. P y =yt pt,y = 1 pt,yt = 1 DP

i |Π | qt,iπi(xt),ℓt E = rt,

y Y wt,ℓy t log|Y| 1 wt,ℓy t |Y| = (1 pt,yt) log|Y| 1 1 pt,yt |Y| +

y =yt (1 pt,y) log|Y| 1 1 pt,y |Y|

(a) (1 pt,yt) log|Y| 1 1 pt,yt |Y| + (|Y| 1)

|Y| 1 log|Y| |Y| 1 1 pt,yt |Y|

(1 pt,yt) log|Y| 1 1 pt,yt |Y| + (1 pt,yt) log|Y| |Y| 1 1 pt,yt |Y|

= (1 pt,yt) log|Y| |Y| 1 (1 pt,yt) 2

(b) rt log|Y| |Y| 1

where step (a) is by applying Jensen s inequality and using 1 pt,y = 1 pt,yt

|Y| 1 , and step (b) is by replacing the expected loss 1 pt,yt by its short-hand notation rt.

Recall that we define the expected cumulative loss as RT = PT t=1 rt. Since when rt [0, 1],

rt log|Y| |Y| 1

r2 t |Y| is concave, we get

y Y wt,ℓy t log|Y| 1 wt,ℓy t |Y| T P rt

T log|Y| |Y| 1 P rt

|Y| = RT log|Y| T 2(|Y| 1)

Since RT is the cumulative loss up to round T, T s incremental rate is no less than RT s incremental rate. Thus, RT T and Tt

Rt+1 . So we get Eq. (14).

Now we are ready to prove Theorem 2.

Proof of Theorem 2. From Lemma 10, we get the following equation as the cumulative query cost

y Y wt,ℓy t log|Y| 1 wt,ℓy t |Y|

Let us assume the expected total loss of best policy is Tµi . From Theorem 1, we get

E [RT ] = E

ln |Π | 2b2 ln 2

Plugging this result into the query complexity bound given by Lemma 11 and Lemma 12, we have

ln |Π | 2b2 ln 2

|Y| log|Y| T 2 (|Y| 1) ln |Π | 1

ln |Π | 2b2 ln 2

ln |Π | 2b2 ln 2

! log|Y| (T|Y|)

ln |Π | 2b2 ln 2

ln |Π | 2b2 ln 2

where γ is defined as Eq. (10) and step (a) by applying c = |Y|.

F Proofs for the Adversarial Setting

In this section, we first prove the regret bound presented in Theorem 3 and then prove the query complexity bound presented in Theorem 4 for Algorithm 1 in the adversarial setting. Lemma 13 builds upon the proof of the hedge algorithm [27], but with an adaptive learning rate.

F.1 Proof of Theorem 3

Lemma 13. Consider the setting of Algorithm 1, Let us define ht,i = exp ηt Lt 1,i i |Π | as exponential cumulative loss of policy i, ηt is the adaptive learning rate and qt is the probability distribution of policies, then

i [|Π |] h T +1,i P i [|Π |] h1,i

i=1 qt,ieℓt,i +

i=1 qt,i eℓt,i 2 .

Proof. We first bound the following term

i [|Π |] ht+1,i P

i [|Π |] ht,i =

i [|Π |] ht,i

i=1 qt,i exp ηteℓt,i

1 ηteℓt,i + η2 t eℓt,i 2

i=1 qt,ieℓt,i + η2 t 2

i=1 qt,i eℓt,i 2 ,

where the inequality is by applying that for x 0, we have ex 1 + x + x2

By taking log on both side, we get

i [|Π |] ht+1,i P

i [|Π |] ht,i log

i=1 qt,ieℓt,i + η2 t 2

i=1 qt,i eℓt,i 2

i=1 qt,ieℓt,i + η2 t 2

i=1 qt,i eℓt,i 2 ,

where step (a) is by applying that log (1 + x) x, when x 1.

Now summing over t = 1 : T yields:

i [|Π |] h T +1,i P

i [|Π |] h1,i =

i [|Π |] ht+1,i P

i [|Π |] ht,i

i=1 qt,ieℓt,i +

i=1 qt,i eℓt,i 2 .

Lemma 14. Consider the setting of Algorithm 1. Let pt,y = P

j [k] I {ˆyt,j = y} wj. The query probability zt satisfies

zt 1 |Y| ln |Y| (pt,yt (1 pt,yt) + pt,y (1 pt,y)) , y = yt.

Proof. We first bound the query probability term

zt = max {δt 0, E (ˆyt, wt)}

= max{δt 0, 1

y Y wt,ℓy t log|Y| 1 wt,ℓy t }

= max{δt 0, 1

y Y (1 pt,y) ln 1 1 pt,y

(a) max{δt 0, 1

y Y (1 pt,y) pt,y 1 ln |Y|}

= max{δt 0, 1 |Y| ln |Y|

y Y (1 pt,y) pt,y}

(b) 1 |Y| ln |Y| (pt,yt (1 pt,yt) + pt,y (1 pt,y)) , y = yt,

where step (a) is by applying ln (1 + x) x 1+x for x > 1,

ln 1 1 pt,y = ln 1 + pt,y 1 pt,y

pt,y 1 pt,y

1 1 pt,y = pt,y,

and where step (b) is by applying a, b R, max {a, b} a.

Proof of Theorem 3. By applying Lemma 13, we got

i [|Π |] h T +1,i P

i [|Π |] h1,i

i=1 qt,ieℓt,i +

i=1 qt,i eℓt,i 2 .

For any policy s, we have a lower bound

P i [|Π |] h T +1,i P

i [|Π |] h1,i log h T +1,s P

i [|Π |] h1,i

(a) = log h T +1,s

= log (n + k) ηT

t=1 eℓt,s, (15)

where step (a) in Eq. (15) is by initializing e L0 = 0, e0 = 1, and P

i [|Π |] h1 = e( ηte L0) = |Π |.

Thus, we have

i=1 qt,ieℓt,i +

i=1 qt,i eℓt,i 2 log (n + k) ηT

i=1 qt,ieℓt,i ηT

t=1 eℓt,s log (n + k) +

i=1 qt,i eℓt,i 2

i=1 qt,ieℓt,i ηT

(b) log (n + k) +

i=1 qt,i eℓt,i 2

i=1 qt,ieℓt,i

(c) log |Π |

i=1 qt,i eℓt,i 2 ,

where step (b) is by applying

i=1 qt,ieℓt,i ηT

i=1 qt,ieℓt,i ηT

and step (c) is by dividing ηT on both side.

Because we have

qt,i eℓt,i 2 = qt,i ET

πi (xt) bℓt 2

P (Ut = 1) πi (xt) ℓt

2 + P (Ut = 0) 0

zt (πi (xt) ℓt)2

zt πi (xt) ℓt

zt πi (xt) ,ℓt ,

it leads to

i=1 qt,ieℓt,i

t=1 eℓt,s log |Π |

zt πi (xt) ,ℓt

(d) log |Π |

η2 t 2 wt,ℓt

where step (d) is by applying P|Π | i=1 qt,i πi (xt) ,ℓt = wt,ℓt .

So we have,

i=1 qt,ieℓt,i

t=1 eℓt,s log |Π |

η2 t 2 wt,ℓt

(e) log |Π |

η2 t 2 1 pt,yt

(f) log |Π |

η2 t 2 1 pt,yt Y0 ((1 pt,yt) pt,yt + (1 pt,y) pt,y)

Y0 pt,yt + 1 pt,y

1 pt,yt pt,y ,

where step (e) is by using wt,ℓt = 1 pt,yt and step (f) by using Lemma 14 and get lower bound of zt as 1 |Y| ln |Y| (pt,yt (1 pt,yt) + pt,y (1 pt,y)) and applying 1 |Y| ln |Y| = Y0.

If pt,yt 1 |Y|,

pt,yt + 1 pt,y

1 pt,yt pt,y 1 |Y|.

If pt,yt < 1 |Y|, y, pt,y 1, δt 1 = 1 maxy,τ [t] pτ,y. Let pt,ˆy = maxy pt,y. Thus, we have wˆy > 1 |Y| and

pt,yt + 1 pt,y

1 pt,yt pt,y pt,yt + wˆy δt 1 1 pt,yt 0 + 1

|Y| δt 1 1 = δt 1 |Y|.

max{pt,yt + pt,y 1 pt,y 1 pt,yt } =

( 1 |Y| if pt,yt 1 |Y|,

δt 1 |Y| if pt,yt < 1 |Y|.

i=1 qt,ieℓt,i

t=1 eℓt,s log |Π |

Y0 pt,yt + 1 wy 1 pt,yt pt,y

(g) log |Π |

max{Y0 δt 1 |Y|, δt 0}

η2 t 2 |Y|2 ln |Y| max{δt 1, δt 0|Y|2 ln |Y|}

(h) log |Π |

η2 t 2 |Y|2 ln |Y|

δt 1+δt 0|Y|2 ln |Y|

t=1 η2 t 1 δt 1 + δt 0|Y|2 ln |Y| |Y|2 ln |Y|,

where step (g) is by getting the lower bound of zt as δt 1 |Y| 1 |Y|, δt 0 δt 0 1 pt,yt and step (h) is by

applying max{A, B} A+B

Let us define ρt minτ [t] δτ 1 = 1 maxc,τ [t] pτ t,y. We get

ET [RT ] log |Π |

t=1 log |Π | 1

T 2 log |Π |

ρt+δt 0|Y|2 ln |Y| |Y|2 ln |Y| q

T , we obtain

ET [RT ] 2 p

|Y|2 ln |Y| p

ρT + δT 0 |Y|2 ln |Y|

T ln |Y| log |Π |

where the last inequality is due to the fact that

ρT + δT 0 |Y|2 ln |Y| > max{ρT , δT 0 } = max{ρT , p

which completes the proof.

F.2 Proof of Theorem 4

Proof of Theorem 4. From Lemma 10, we get the following equation as the cumulative query cost

P y Y wt,ℓy t log|Y| 1 wt,ℓy t |Y|

Let us assume the expected total loss of best policy is LT, . Thus, from Theorem 3, we get the expected cumulative loss

E [RT ] = E

T ln |Y| log |Π |

1/T} + LT, .

Now plugging the regret bound RT proved in Theorem 3 into the query complexity bound given by Lemma 12, we have

y Y wt,ℓy t log|Y| 1 wt,ℓy t |Y|

T ln |Y| log |Π |

1/T } + LT,

log|Y| T 2(|Y| 1) 2|Y| r

T ln |Y| log |Π |

1/T } + LT,

T ln |Y| log |Π |

1/T } + LT,

log|Y| T|Y|

T ln |Y| log |Π |

1/T } + LT,

log|Y| T + 1

Finally, by applying query complexity upper bound of Lemma 11, we got

T ln |Y| log |Π |

1/T } + LT,

log|Y| T + 1

Since the second term on the RHS dominates the upper bound, we have

1/T } + LT,

where step (a) is obtained by suppressing constant coefficients involving |Y| into the O notation.

G Additional Experiments

In this section, we further evaluate CAMS and provide additional experimental results (complementary to the main results presented in Fig. 2) under the following scenarios:

1. In App. G.1, we demonstrate that CAMS outperforms the baselines on a large scale dataset as well. 2. In App. G.2, we perform ablation study of three query strategies CAMS (entropy), variance and random strategy. CAMS (Entropy) achieves the minimum cumulative loss for CIFAR10, DRIFT, and VERTEBRAL under the same query cost and outperform the other query strategies. 3. In a mixture of experts environment, CAMS converges to the best policy and outperforms all others (App. G.3); 4. In a non-contextual (no experts) environment, CAMS has approximately equal performance as Model Picker to reach the best classifier effectively (App. G.4); 5. In an adversarial environment, CAMS can efficiently recover from the adversary and approach the performance of the best classifier (App. G.5); 6. In a complete sub-optimal expert environment, a variant of the CAMS algorithm, namely CAMS-MAX, which deterministically picks the most probable policy and selects the most probable model, outperforms CAMS-Random-Policy, which randomly samples a policy and selects the most probable model (App. G.7 & App. G.8). However, CAMS-MAX at most approaches the performance of the best policy. In contrast, perhaps surprisingly, CAMS is able to outperform the best policy on both VERTEBRAL and HIV (App. G.6). 7. In App. G.9, we summarize the maximum query cost under a fixed number of realizations with its associated cumulative loss for all baselines (exclude oracle) on all benchmarks in experiment section. 8. In App. G.10, we compare the query complexity of each baselines and demonstrate that CAMS has the lowest query cost increasing rate on CIFAR10, DRIFT and VERTEBRAL dataset. 9. In previous studies, we assume that the data comes in an online format. In App. G.11, we assume we know the data stream length ahead and applying the scaling parameter to each algorithm to query their top data points from hindsight. CAMS still outperforms all the baselines. 10. In App. G.12, we compare CAMS with CAMS-nonactive, a greedy version (query label for each incoming data point). Although CAMS query much less data, it still performs equally well or even better than the greedy version. 11. In App. G.13, we demonstrates that CAMS can achieve negative RCL on all benchmarks, which means it outperforms any algorithms that chase the best classifier where the horizontal 0 line represents the performance benchmark of best classifier.

G.1 Performance of CAMS at scale: Experimental results on Cov Type

We scaled up our experiments on a larger dataset, Cov Type [24]. The Cov Type dataset offers details about different types of forest cover in the United States. It contains details including slope, aspect, elevation, measurements of the wilderness area, and the type of forest cover. Cov Type has 580K samples, of which 100K instances were chosen at random as online stream for testing. Fig. 4 demonstrated that CAMS outperforms all baselines which is consistent with the existing results in experiment section.

G.2 Query strategies ablation comparison

Using the same CAMS model recommendation section, we compare three query strategies: the adaptive model-disagreement-based query strategy in Line 10-14 of Fig. 1 (referred to as entropy in the following), the variance-based query strategy from Model Picker [39] (referred to as variance), and a random query strategy. Fig. 5 shows that CAMS s adaptive query strategy has the sharpest converge rate on cumulative loss, which demonstrates the effectiveness of the queried labels. Moreover, entropy achieves the minimum cumulative loss for CIFAR10, DRIFT, and VERTEBRAL under the same query cost. For the HIV dataset, there is no clear winner between entropy and variance since the mean of their performance lie within the error bar of each other for the most part.

Relative cumulative loss

0 20000 40000 60000 80000 100000 Round

(a) Relative Cumulative Loss

Number of queries

0 20000 40000 60000 80000 100000 Round

Cumulative loss

0 1000 2000 3000 4000 5000 Query cost

(c) Cumulative Loss VS Cost

Figure 4: Comparing CAMS with 7 baselines on Cov Type in terms of relative cumulative loss, query complexity, and cost effectiveness. CAMS outperforms all baselines. (Left) Performance measured by relative cumulative loss (i.e. loss against the best classifier) under a fixed query cost B (where B = 1000). (Middle) Number of queries and (Right) Performance of cumulative loss by increasing the query cost, for a fixed number of rounds T (where T = 100, 000) and maximal query cost B (where B = 5000 ). Algorithms: 4 contextual {Oracle, CQBC, CIWAL, CAMS} and 4 non-contextual baselines {RS, QBC, IWAL, MP} are included (see Section ). 90% confident interval are indicated in shades.

variance random entropy

Cumulative loss

0 200 400 600 800 1000 1200 Query cost

(a) CIFAR10

0 50 100 150 200 250 300 Query cost

0 10 20 30 40 50 60 70 80 Query cost

(c) VERTEBRAL

0 250 500 750 1000 1250 1500 1750 Query cost

Figure 5: Ablation study of three query strategies (entropy, variance, random) for 4 diverse benchmarks based on the same model recommendation strategy. Under the same query cost constraint, CAMS s entropy-based strategy exceeds the performance of the other two strategies on non-binary benchmarks in terms of query cost and cumulative lost. 90% confident intervals are indicated in shades.

G.3 Comparing CAMS with each individual expert

We evaluate CAMS by comparing it with all the policies available in various benchmarks. The policies in each benchmark are summarized in App. D.2 and Table D.5. The empirical results in Fig. 6 demonstrate that CAMS could efficiently outperform all policies and converge to the performance of the best policy with only slight increase in query cost in all benchmarks. In particular, on the VERTEBRAL and HIV benchmarks, CAMS even outperforms the best policy.

G.4 Comparing CAMS against Model Picker in a context-free environment

CAMS outperforms Model Picker in Fig. 2, by leveraging the context information for adaptive model selection. In a context-free environment, Π = { }, so Π := {πconst 1 , . . . , πconst k }, where πconst j ( ) := ej represents a policy that only recommends a fixed model. In this case, selecting the best policy to CAMS equals selecting the best single model. Fig. 7 demonstrates that the mean of CAMS and Model Picker lies in the shades of each other, which means CAMS has approximately the same performance as model picker considering the randomness on all benchmarks.

G.5 Robustness against malicious experts in adversarial environments

When given only malicious and random advice policies, the conventional contextual online learning from experts advice framework will be trapped in the malicious or random advice. In contrast, CAMS could efficiently identify these policies and avoid taking advice from them. Meanwhile, it also successfully identifies the best classifier to learn to reach its best performance.

best policy top sub-optimal policies CAMS

Cumulative loss

0 50 100 150 200 250 300 350 Query cost

(a) CIFAR10

0 50 100 150 200 250 Query cost

5 10 15 20 25 30 35 Query cost

(c) VERTEBRAL

0 100 200 300 400 500 600 700 800 Query cost

Figure 6: Comparing CAMS with every single policy (only plotted top performance policies in Figure). CAMS could approach the best expert and exceed all others with limited queries. In particular, on VERTEBRAL and HIV Benchmarks, CAMS outperforms the best expert. 90% confident intervals are indicated in shades.

MP CAMS best classifier top sub-optimal classifiers

Cumulative loss

0 200 400 600 800 1000 1200 Query cost

MP CAMS best classifier sub-optimal classifiers

(a) CIFAR10

0 50 100 150 200 250 300 350 400 Query cost

5 10 15 20 25 30 Query cost

(c) VERTEBRAL

20 40 60 80 100 120 140 Query cost

Figure 7: Comparing the model selection strategy of CAMS and Model Picker baseline based on the same variance-based query strategy in a context-free environment. CAMS has approximately the same performance as Model Picker on all the benchmarks. 90% confident intervals are indicated in shades.

The novelty in CAMS that enables this robustness is that we add the constant policies {πconst 1 , . . . , πconst k } into the policy set Π to form the new set as Π . To illustrate the performance difference, we have created a variant of CAMS by adapting to the conventional approach (named CAMS-conventional). Fig. 8 demonstrates that CAMS could outperform all the malicious and random policies and converge to the performance of the best classifier. CAMS-conventional: We create the CAMS-conventional algorithm as the CAMS using policy set Π, not Π .

best classifier top sub-optimal policies CAMS CAMS-conventional

Cumulative loss

0 50 100 150 200 250 300 350 400 Query cost

(a) CIFAR10

0 50 100 150 200 250 Query cost

5 10 15 20 25 30 35 Query cost

(c) VERTEBRAL

0 250 500 750 1000 1250 1500 1750 Query cost

Figure 8: Evaluating the robustness of CAMS compared to the conventional learning from experts advice (CAMS-conventional) in a complete malicious and random policies environment. When no good policy is available, CAMS could recover from malicious advice and successfully approach the performance of the best classifier. In contrast, the conventional approach will be trapped in malicious advice. 90% confident intervals are indicated in shades.

G.6 Outperformance over the best policy/expert

We also observe that CAMS does not stop at approaching the best policy or classifier performance. Sometimes, it even outperforms all the policies and classifiers, and Fig. 9 demonstrates such a case. To demonstrate the advantage of CAMS, we create two variant versions of CAMS: (1) CAMS-MAX (App. G.7), (2) CAMS-Random-Policy (App. G.8). CAMS-MAX and CAMS-Random-Policy use the same algorithm as CAMS in adversarial settings but have different model selection strategies for ablation study in the stochastic settings.

We evaluate the three algorithms on VERTEBRAL and HIV benchmarks in terms of (a) normal policies (Fig. 9 Left), (b) classifiers (Fig. 9 Middle), and (c) malicious and random policies (Fig. 9 Right). In the normal policies column, we only compare the policies with regular policies giving helpful advice. In the classifier column, we compare them with the performance of classifiers only. In the malicious and random policies column, we compare them with unreasonable policies only.

Fig. 9 demonstrates that all three algorithms could outperform the malicious/random policies. However, CAMS-Random-Policy does not outperform the best classifier while both CAMS and CAMSMAX can on both benchmarks. CAMS-MAX approaches the performance of the best policy but does not outperform the best policy on both benchmarks. Finally, perhaps surprisingly, CAMS outperforms the best policy (Oracle) on both benchmarks and continues to approach the hypothetical, optimal policy (with 0 cumulative loss).

This surprising factor is contributed by the adaptive weighted policy of CAMS, which adaptively creates a better policy by combining the advantage of each sub-optimal policy and classifier to reach the performance of the hypothetical, optimal policy (defined as PT t=1 mini [n+k] eℓt,i). The second reason could be that the benchmark we created, or any real-world cases, will not be strictly in a stochastic setting (in which a single policy outperforms all others or has lower µ in every round). The weight policy strategy can make a better combination of advice for this case.

best classifier best policy CAMS_MAX CAMS_random_policy CAMS top suboptimal policies/classifiers

Cumulative loss

5 10 15 20 25 30 35

5 10 15 20 25 30 35

5 10 15 20 25 30 35

Cumulative loss

0 100 200 300 400 500 600 700 800 Query cost

(a) normal policies

0 200 400 600 800 1000 1200 1400 Query cost

(b) classifiers

0 200 400 600 800 1000 1200 1400 Query cost

(c) malicious and random policies

Figure 9: Comparing CAMS, CAMS-MAX and CAMS-RANDOM-POLICY with top policies and classifiers in the VERTEBRA and HIV benchmarks. They outperform all the malicious/random policies. Moreover, CAMS and CAMS-MAX outperform the best classifier. Finally, only CAMS even exceeds the best policy (Oracle) in both benchmarks and continues approaching the hypothetical, optimal policy (0 cumulative loss). 90% confident intervals are indicated in shades.

G.7 The CAMS-MAX algorithm

CAMS-MAX is a variant of CAMS. In an adversarial setting, they share the same algorithm. However, in a stochastic setting, CAMS-MAX gets the index i of max value in the probability distribution of policy q, and selects the model with the max value in πi (xt) to recommendation. The difference is marked as blue color in Fig. 10.

1: Input: Models F, policies Π , #rounds T, budget b 2: Initialize loss L0 0; query cost C0 0 3: for t = 1, 2, ..., T do 4: Receive xt 5: ηt SETRATE(t, xt, |Π |)

6: Set qt,i exp ηt Lt 1,i i |Π |

7: jt RECOMMEND(xt, qt) 8: Output ˆyt,jt ft,jt as the prediction for xt 9: Compute zt in Eq. (4) 10: Sample Ut Ber (zt) 11: if Ut = 1 and Ct b then 12: Query the label yt 13: Ct Ct 1 + 1 14: Compute ℓt: ℓt,j = I {ˆyt,j = yt} , j [|F|]

15: Estimate model loss: ˆℓt,j = ℓt,j

zt , j [|F|]

16: ℓt: ℓt,i πi(xt), ˆℓt,j , i [|Π |] 17: Lt = Lt 1 + ℓt 18: else 19: Lt = Lt 1 20: Ct Ct 1 21: end if 22: end for

21: procedure SETRATE(t, xt, m) 22: if STOCHASTIC then

t 24: end if 25: if ADVERSARIAL then 26: Set ρt as in adversarial section

t + ρt c2 ln c q

T 28: end if 29: return ηt 30: end procedure

29: procedure RECOMMEND(xt, qt) 30: if STOCHASTIC then 31: it maxind(qt) 32: jt maxind(πit (xt)) 33: end if 34: if ADVERSARIAL then 35: it qt 36: jt πit (xt) 37: end if 38: return jt 39: end procedure

Figure 10: The CAMS-MAX Algorithm

G.8 The CAMS-Random-Policy algorithm

1: Input: Models F, policies Π , #rounds T, budget b 2: Initialize loss L0 0; query cost C0 0 3: for t = 1, 2, ..., T do 4: Receive xt 5: ηt SETRATE(t, xt, |Π |)

6: Set qt,i exp ηt Lt 1,i i |Π |

7: jt RECOMMEND(xt, qt) 8: Output ˆyt,jt ft,jt as the prediction for xt 9: Compute zt in Eq. (4) 10: Sample Ut Ber (zt) 11: if Ut = 1 and Ct b then 12: Query the label yt 13: Ct Ct 1 + 1 14: Compute ℓt: ℓt,j = I {ˆyt,j = yt} , j [|F|]

15: Estimate model loss: ˆℓt,j = ℓt,j

zt , j [|F|]

16: ℓt: ℓt,i πi(xt), ˆℓt,j , i [|Π |] 17: Lt = Lt 1 + ℓt 18: else 19: Lt = Lt 1 20: Ct Ct 1 21: end if 22: end for

21: procedure SETRATE(t, xt, m) 22: if STOCHASTIC then

t 24: end if 25: if ADVERSARIAL then 26: Set ρt as in adversarial section

t + ρt c2 ln c q

T 28: end if 29: return ηt 30: end procedure

29: procedure RECOMMEND(xt, qt) 30: if STOCHASTIC then 31: it qt 32: jt maxind(πit (xt)) 33: end if 34: if ADVERSARIAL then 35: it qt 36: jt πit (xt) 37: end if 38: return jt 39: end procedure

Figure 11: The CAMS-Random-Policy Algorithm

CAMS-Random-Policy is a variant of CAMS. It shares the same algorithm with CAMS in an adversarial environment. However, it uses a random sampling policy method in a stochastic setting.

It randomly samples the policy from the probability distribution of policy q, and selects the model with max value in πi (xt) to recommendation. The difference is marked as blue color in Fig. 11.

G.9 Maximal queries from experiments

Table 6 in this section summarizes the maximum query cost for a given data stream (of fixed total size), with its associated cumulative loss for all baselines (exclude Oracle) on all benchmarks in experiment section. The result in this table is slightly different from the query complexity curves of Fig. 2 (Middle). The curve in Fig. 2 (Middle) takes the average value, while the table takes the maximal value from a fixed number of simulations. CAMS wins over all baselines (other than Oracle) in terms of query cost on CIFAR10, DRIFT, and VERTEBRAL benchmarks. CAMS outperforms all baselines in terms of cumulative loss on DRIFT, VERTEBRAL, and HIV benchmarks. In particular, CAMS outperforms both cumulative loss and query cost on the DRIFT and VERTEBRAL benchmarks.

Algorithm CIFAR10 DRIFT VERTEBRAL HIV

Max queries, Cumulative loss 1200, 10000 2000, 3000 80, 80 2000, 4000 RS 1200, 2916 2000, 766 80, 19 2000, 143 QBC 1200, 2857 1904, 771 72, 20 2000, 139 IWAL 1200, 2854 2000, 760 80, 19 690, 140 MP 1200, 2885 493, 803 33, 25 153, 148 CQBC 1200, 2284 1900, 744 68, 13 2000, 124 CIWAL 1200, 2316 2000, 746 80, 12 690, 124 CAMS 348, 2348 251, 710 32, 11 782, 112 Table 5: Maximal queries from experiments

G.10 Query complexity

To achieve the same level of prediction accuracy (measured by average cumulative loss over a fixed number of rounds), CAMS incurs less than 10% of the label cost of the best competing baselines on CIFAR10 (10K examples), and 68% the cost on VERTEBRAL (see Fig. 12); Fig. 12 8 and Table 6 also demonstrate the compelling effectiveness of CAMS s query strategy outperforming all baselines in terms of query cost in VERTEBRAL, DRIFT, and CIFAR10 benchmarks, which is consistent with our query complexity bound in Theorem 2.

Number of queries

0 2000 4000 6000 8000 10000 Round

(a) CIFAR10

0 500 1000 1500 2000 2500 3000 Round

0 10 20 30 40 50 60 70 80 Round

(c) VERTEBRAL

0 500 1000 1500 2000 2500 3000 3500 4000

Figure 12: Comparing CAMS with 7 baselines on 4 diverse benchmarks in terms of query complexity (Number of queries). CAMS outperforms all baselines for a fixed number of rounds T (where T = 10000, 3000, 80, 4000 from left to right) and maximal query cost B (where B = 1200, 2000, 80, 2000 from left to right). Algorithms: 4 contextual {Oracle, CQBC, CIWAL, CAMS} and 4 non-contextual baselines {RS, QBC, IWAL, MP} are included (see Section ). 90% confident interval are indicated in shades.

G.11 Fine-tuning the query probabilities for stochastic streams

For the experimental results we reported in the main paper, we consider a streaming setting where the data arrives online in an arbitrary order and arbitrary length. Therefore, for both CAMS and

8We also consider variants for each algorithm (other than Random and Oracle) where we scale the query probabilities based on the early-phase performance and observe similar behavior. See App. G.11 for the corresponding results.

the baselines, we used the exact off-the-shelf query criteria as described in experiment setup section without fine-tuning the query probabilities, which could be otherwise desirable in certain scenarios (e.g. for stochastic streams, where the query probability can be further optimized).

In this section, we consider such scenarios, and conduct an additional set of experiments to further demonstrate the performance of CAMS assuming stochastic data streams. Given the stream length T and query budget b, we may optimize each algorithm by scaling their query probabilities, so that each algorithm allocates its query budget to the top b informative labels in the entire online stream based on its own query criterion. Note that in practice, finding the exact scaling parameter is infeasible, as we do not know the online performance unless we observe the entire data stream. While it is challenging to determine the scaling factor for each algorithm under the adversarial setting, one can effectively estimate the scaling factor for stochastic streams, where the context arrives i.i.d..

Concretely, we use the early budget to decide the scaling parameter in our following evaluation: Firstly, we use a small fraction (i.e. T/10) of the online stream and see how much queries bearly each algorithm consumed. Then we calculate the scaling parameter s = (b bearly)

T T/10 T/10

bearly and multiply the scaling factor with the query probability of each algorithm for the remaining 9 10 T rounds. The results in Fig. 13 demonstrate that CAMS still outperforms all the baselines (excluding Oracle) when all algorithms select the top b data of the whole online stream to query. The improvement of CAMS over the baseline approaches does not differ much between the two versions (with or without scaling) of the experiments as shown in the bottom plots of Fig. 2 and Fig. 13.

For a head-to-head comparison between the bottom plots of Fig. 2 and Fig. 13, note that the total number of rounds stays the same for DRIFT (T = 3000), VERTEBRAL (T = 80), and HIV (T = 4000); while we used half the rounds and half the maximal budget for CIFAR10 (T = 5000) for the version with scaling. Roughly speaking, the cumulative regret plots for the baselines were "streched out" to cover the full allocated budget after scaling, but we do not observe a significant difference in terms of the absolute gain in terms of the cumulative loss. Another way to read the difference between the two plots is to compare the cumulative losses at the budget range where all algorithms were not cut off early: e.g., for DRIFT, when Query Cost is 250, the cumulative losses for the competing algorithm stay roughly the same under the two evaluation scenarios.

RS Oracle QBC IWAL MP CQBC CIWAL CAMS

Cumulative loss

0 100 200 300 400 500 600 Query cost

(a) CIFAR10

0 100 200 300 400 500 600 700 800 Query cost

0 10 20 30 40 50 60 Query cost

(c) VERTEBRAL

0 100 200 300 400 500 600 Query cost

Figure 13: Comparing CAMS with 7 model selection baselines on 4 diverse benchmarks in terms of cost effectiveness after applying the scaling parameter to each algorithm. CAMS outperforms all baselines (excluding Oracle). Performance of cumulative loss by increasing the query cost, for a fixed number of rounds T (where T = 5000, 3000, 80, 4000 from left to right) and maximal query cost B (where B = 600, 800, 60, 600 from left to right). Algorithms: 4 contextual {Oracle, CQBC, CIWAL, CAMS} and 4 non-contextual baselines {RS, QBC, IWAL, MP} are included (see Section ). 90% confident interval are indicated in shades.

G.12 Ablation study on the active query strategy

In this section, we compare the performance of CAMS and its non-active variant (CAMS-nonactive), which queries the label for each incoming data point. As shown in Fig. 14, CAMS performs equally well or better than CAMS-nonactive, even though it queries significantly less data. Surprisingly, on the DRIFT dataset, CAMS significantly outperforms CAMS-nonactive, even when using less than 10 percent of the query budget (Fig. 14b). This demonstrates that CAMS selectively choose the data to query to maximal optimize policy improvement, while CAMS queries all data points, regardless of their usefulness or noise, which hampers policy improvement and convergence.

CAMS CAMS-nonactive

Number of queries

0 1000 2000 3000 4000 5000 Round

0 500 1000 1500 2000 2500 3000 Round

0 10 20 30 40 50 60 70 80 Round

0 500 1000 1500 2000 2500 3000 3500 4000

Cumulative loss

0 1000 2000 3000 4000 5000 Query cost

(a) CIFAR10

0 500 1000 1500 2000 2500 3000 Query cost

0 10 20 30 40 50 60 70 80 Query cost

(c) VERTEBRAL

0 500 1000 1500 2000 2500 3000 3500 Query cost

Figure 14: Comparing CAMS (in red) with CAMS-nonactive (in blue) on 4 diverse benchmarks in terms of query complexity, and cost effectiveness. CAMS outperforms or performs equally well to CAMS-nonactive with much less queried labels for all benchmarks. (Top) Number of queries and (Bottom) Performance of cumulative loss by increasing the query cost, for a fixed number of rounds T (where T = 5000, 3000, 80, 4000 from left to right) and maximal query cost B (where B = T = 5000, 3000, 80, 4000 from left to right). 90% confident interval are indicated in shades.

G.13 Relative Cumulative Loss

Relative cumulative loss (RCL). At round t, we define RCL as Lt,ji Lt,j , where Lt,j stands for the cumulative loss (CL) of the policy always selecting the best classifier, and Lt,ji stands for the CL of policy i.

Relative cumulative loss

0 2000 4000 6000 8000 10000 Round

(a) CIFAR10

0 500 1000 1500 2000 2500 3000 Round

0 10 20 30 40 50 60 70 80 Round

(c) VERTEBRAL

0 500 1000 1500 2000 2500 3000 3500 4000

Figure 15: Comparing CAMS with 7 baselines on 4 diverse benchmarks in terms of loss trajectory. CAMS outperforms all baselines. Performance measured by relative cumulative loss (i.e. loss against the best classifier) under a fixed query cost B (where B = 200, 400, 30, 400 from left to right). Algorithms: 4 contextual {Oracle, CQBC, CIWAL, CAMS} and 4 non-contextual baselines {RS, QBC, IWAL, MP} are included (see Section ). 90% confident interval are indicated in shades.

The RCL under the same query cost for all baselines is shown in Fig. 15. The loss trajectory demonstrates that CAMS efficiently adapts to the best policy after only a few rounds and outperforms all baselines in all benchmarks. The result also demonstrates that CAMS can achieve negative RCL on all benchmarks, which means it outperforms any algorithms that chase the best classifier, as the horizontal 0 line represents the performance benchmark of best classifier. This empirical result aligns with Theorem 1 that, in the worst scenario, if the best classifier is the best policy, CAMS will achieve its performance. Otherwise, CAMS will reach a better policy and incurs no regret.

CAMS could achieve such performance because when an Oracle fails to achieve 0 loss over all instances and contexts, CAMS has the opportunity to outperform the Oracle in those rounds Oracle does not make the best recommendation. For instance, the stochastic version of CAMS (Line 22-23; Line 30-32 in Fig. 1) may achieve this by recommending a model using the weighted majority vote among all policies. Therefore, one can view CAMS as adaptively constructing a new policy at each

round by combining the advantages of each sub-optimal policy, which may outperform any single expert/policy. Furthermore, for the experiments we ran (or in most real-world scenarios), the data streams are not strictly in a stochastic setting (in which a single policy outperforms all others or has a lower expected loss in every round). The weighted policy strategy may find a better combination of "advices" in such cases (see Fig. 2 and App. G.6).

Global Rebuttal Response

H Experiments on Image Net

0 500 1000 1500 2000 2500 Query cost

Cumulative loss

RS Oracle QBC IWAL MP CQBC CIWAL CAMS

(a) IMAGENET

Figure 16: Comparison of CAMS with 7 baselines on IMAGENET benchmark in terms of cost effectiveness. We plot the cumulative loss as we increase the query cost for a fixed number of rounds T and maximal query cost B (T = 3000, and B = 2500). CAMS outperforms all baselines. Algorithms: 4 contextual {Oracle, CQBC, CIWAL, CAMS} and 4 non-contextual baselines {RS, QBC, IWAL, MP} are included. 90% confident interval are indicated in shades.

I Comparing CAMS against recent works in active learning

Active Learning Setting / Algorithms Coreset Batch-BALD BADGE; VAAL; Cluster Margin

BALANCE; GLISTER Ve SSAL Model Picker CAMS

Streaming, sequential Streaming, batch Pool-based, batch

Table 6: Selective comparison against recent works in active learning. Among these algorithms, Coreset (Sener & Savarese, 2017) is a diversity sampling strategy for deep active learning; Batch BALD is an uncertainty sampling strategy; BADGE (Ash et al., 2019), VAAL (Sinha et al., 2019) Cluster Margin (Citovsky et al., 2021), and Ve SSAL (Saran et al., 2023) represent strategies that combine both; GLISTER (Killamsetty et al., 2020) and BALANCE (Zhang et al., 2023) represent decision-theoretic approaches that directly optimize the utility of queries.

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Guidelines:

The answer NA means that the paper does not include experiments. The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) The assumptions made should be given (e.g., Normally distributed errors). It should be clear whether the error bar is the standard deviation or the standard error of the mean. It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? Answer: [Yes] Justification: It can be found at the supplementary Appendix D.6. Guidelines:

The answer NA means that the paper does not include experiments. The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn t make it into the paper). 9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the Neur IPS Code of Ethics https://neurips.cc/public/Ethics Guidelines? Answer: [Yes] Justification: The codes are anonymous as well. Guidelines:

The answer NA means that the authors have not reviewed the Neur IPS Code of Ethics. If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 10. Broader Impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? Answer: [Yes] Justification: It has been discussed at Abstract and Section 1. Guidelines:

The answer NA means that there is no societal impact of the work performed. If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? Answer: [NA] Justification: Guidelines:

The answer NA means that the paper poses no such risks. Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. 12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [NA] Justification: Guidelines:

The answer NA means that the paper does not use existing assets. The authors should cite the original paper that produced the code package or dataset. The authors should state which version of the asset is used and, if possible, include a URL. The name of the license (e.g., CC-BY 4.0) should be included for each asset. For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. If this information is not available online, the authors are encouraged to reach out to the asset s creators. 13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? Answer: [NA] Justification: Guidelines:

The answer NA means that the paper does not release new assets. Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. The paper should discuss whether and how consent was obtained from people whose asset is used. At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file. 14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? Answer: [NA] Justification: Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. According to the Neur IPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? Answer: [NA] Justification: Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the Neur IPS Code of Ethics and the guidelines for their institution. For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.